Geometry Theorems

Why This Matters

Geometry theorems aren't just formulas to memorize—they're the logical tools that unlock nearly every shape-based problem on the PSAT. You're being tested on your ability to recognize which theorem applies to a given situation and how to set up the equation correctly. The PSAT loves to combine these concepts: a problem might require you to find a missing angle using the Triangle Sum Theorem, then apply the Pythagorean Theorem to find a side length.

The key to success is understanding the underlying principles: angle relationships, proportional reasoning, and properties of congruence and similarity. When you know why these theorems work, you can spot opportunities to use them even when a problem doesn't explicitly tell you which one to apply. Don't just memorize the formulas—know what geometric principle each theorem demonstrates and when it's your go-to tool.

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Right Triangle Relationships

Right triangles are the foundation of PSAT geometry. Their special properties—particularly the 90° angle—create predictable relationships between sides and angles that appear in countless problem types.

Pythagorean Theorem

• $a^2 + b^2 = c^2$—where $c$ is always the hypotenuse (the side opposite the right angle)
• Common Pythagorean triples like 3-4-5, 5-12-13, and 8-15-17 save calculation time on test day
• Applies only to right triangles—if you don't have a 90° angle confirmed, you can't use this theorem

Triangle Sum Theorem

• Interior angles sum to exactly 180°—use this to find any missing angle when two are known
• Works for all triangles, not just right triangles, making it one of the most versatile tools
• Combine with other theorems—once you find all angles, you can identify special triangles or apply similarity rules

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Angle Relationships with Parallel Lines

When parallel lines are cut by a transversal, they create a predictable pattern of equal and supplementary angles. Recognizing these patterns quickly is essential for solving multi-step angle problems.

Parallel Lines and Transversals Theorem

• Corresponding angles are equal—same position at each intersection (think: both above the parallel line, both on the right of the transversal)
• Alternate interior angles are equal—opposite sides of the transversal, between the parallel lines
• Consecutive interior angles are supplementary—same side of the transversal, between parallels, summing to 180°

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Triangle Congruence and Similarity

These theorems help you prove that triangles are either identical (congruent) or proportionally scaled (similar). Understanding the difference is crucial—congruence means equal measurements, while similarity means equal ratios.

Congruent Triangle Theorems (SSS, SAS, ASA, AAS)

• SSS (Side-Side-Side)—three pairs of equal sides guarantee congruent triangles
• SAS (Side-Angle-Side)—two sides and the included angle (the angle between those sides) must match
• ASA and AAS—two angles plus one side; for ASA the side is between the angles, for AAS it's not

Similar Triangle Theorems (AA, SAS, SSS)

• AA (Angle-Angle)—two equal angles guarantee similarity (the third angle must also be equal by Triangle Sum Theorem)
• SAS Similarity—two sides in proportion with equal included angles proves similarity
• SSS Similarity—all three sides in the same ratio confirms proportional triangles

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Proportional Segments in Triangles

These theorems reveal how specific lines within triangles create proportional relationships. They're particularly useful when a problem involves midpoints, angle bisectors, or segment ratios.

Midpoint Theorem

• Midsegment is parallel to the third side—connecting midpoints of two sides creates a segment with special properties
• Midsegment length equals half the third side—if the base is 10, the midsegment is 5
• Creates similar triangles—the smaller triangle formed is similar to the original with a 1:2 ratio

Angle Bisector Theorem

• Divides the opposite side proportionally—the ratio $\frac{BD}{DC} = \frac{AB}{AC}$ when AD bisects angle A
• Adjacent sides determine the proportion—larger adjacent side corresponds to larger segment on the opposite side
• Set up proportions to solve—cross-multiply to find unknown segment lengths

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Circle Theorems

Circle problems on the PSAT often combine angle relationships with segment lengths. The key is recognizing whether you're dealing with inscribed angles, central angles, or external points.

Inscribed Angle Theorem

• Inscribed angle equals half the intercepted arc—an angle with its vertex on the circle measures half the arc it "sees"
• Central angle equals the full arc—compare this to inscribed angles for problems involving both
• Inscribed angles intercepting the same arc are equal—useful for proving angle relationships in circles

Tangent-Secant Theorem

• $(\text{tangent})^2 = (\text{whole secant}) \times (\text{external part})$—when both are drawn from the same external point
• Tangent line is perpendicular to the radius—at the point of tangency, creating a right angle you can use
• External point problems—look for this setup when a problem shows lines drawn from outside a circle

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Area and Volume Formulas

These formulas appear frequently on the PSAT, often combined with other theorems. Memorize them cold—you'll need to apply them quickly.

Area and Volume Formulas for Common Shapes

• 2D areas: Rectangle $A = lw$, Triangle $A = \frac{1}{2}bh$, Circle $A = \pi r^2$
• 3D volumes: Rectangular prism $V = lwh$, Cylinder $V = \pi r^2 h$
• Circle circumference $C = 2\pi r$—often needed when problems involve arc length or circular motion

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Quick Reference Table

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Self-Check Questions

1. Which two theorems would you use together to find a missing side length when given two angles and one side of a right triangle?

2. How do you distinguish between when to use congruence theorems versus similarity theorems on a PSAT problem?

3. Compare and contrast the Midpoint Theorem and the Angle Bisector Theorem—what type of proportion does each create?

4. If an inscribed angle measures 35°, what is the measure of its intercepted arc? What theorem did you use?

5. A problem shows two parallel lines cut by a transversal and asks you to find an unknown angle. What angle relationships should you check first, and how do you know if angles are equal or supplementary?